Optimal. Leaf size=133 \[ -\frac {\left (a^2-b^2 x^2\right )^{3/2}}{21 a^2 b (a+b x)^5}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{9 a b (a+b x)^6}-\frac {2 \left (a^2-b^2 x^2\right )^{3/2}}{315 a^4 b (a+b x)^3}-\frac {2 \left (a^2-b^2 x^2\right )^{3/2}}{105 a^3 b (a+b x)^4} \]
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Rubi [A] time = 0.05, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {659, 651} \begin {gather*} -\frac {2 \left (a^2-b^2 x^2\right )^{3/2}}{315 a^4 b (a+b x)^3}-\frac {2 \left (a^2-b^2 x^2\right )^{3/2}}{105 a^3 b (a+b x)^4}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{21 a^2 b (a+b x)^5}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{9 a b (a+b x)^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 651
Rule 659
Rubi steps
\begin {align*} \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^6} \, dx &=-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{9 a b (a+b x)^6}+\frac {\int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^5} \, dx}{3 a}\\ &=-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{9 a b (a+b x)^6}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{21 a^2 b (a+b x)^5}+\frac {2 \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^4} \, dx}{21 a^2}\\ &=-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{9 a b (a+b x)^6}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{21 a^2 b (a+b x)^5}-\frac {2 \left (a^2-b^2 x^2\right )^{3/2}}{105 a^3 b (a+b x)^4}+\frac {2 \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^3} \, dx}{105 a^3}\\ &=-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{9 a b (a+b x)^6}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{21 a^2 b (a+b x)^5}-\frac {2 \left (a^2-b^2 x^2\right )^{3/2}}{105 a^3 b (a+b x)^4}-\frac {2 \left (a^2-b^2 x^2\right )^{3/2}}{315 a^4 b (a+b x)^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 74, normalized size = 0.56 \begin {gather*} \frac {\sqrt {a^2-b^2 x^2} \left (-58 a^4+25 a^3 b x+21 a^2 b^2 x^2+10 a b^3 x^3+2 b^4 x^4\right )}{315 a^4 b (a+b x)^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.59, size = 74, normalized size = 0.56 \begin {gather*} \frac {\sqrt {a^2-b^2 x^2} \left (-58 a^4+25 a^3 b x+21 a^2 b^2 x^2+10 a b^3 x^3+2 b^4 x^4\right )}{315 a^4 b (a+b x)^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 171, normalized size = 1.29 \begin {gather*} -\frac {58 \, b^{5} x^{5} + 290 \, a b^{4} x^{4} + 580 \, a^{2} b^{3} x^{3} + 580 \, a^{3} b^{2} x^{2} + 290 \, a^{4} b x + 58 \, a^{5} - {\left (2 \, b^{4} x^{4} + 10 \, a b^{3} x^{3} + 21 \, a^{2} b^{2} x^{2} + 25 \, a^{3} b x - 58 \, a^{4}\right )} \sqrt {-b^{2} x^{2} + a^{2}}}{315 \, {\left (a^{4} b^{6} x^{5} + 5 \, a^{5} b^{5} x^{4} + 10 \, a^{6} b^{4} x^{3} + 10 \, a^{7} b^{3} x^{2} + 5 \, a^{8} b^{2} x + a^{9} b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.24, size = 289, normalized size = 2.17 \begin {gather*} \frac {2 \, {\left (\frac {207 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}}{b^{2} x} + \frac {1143 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{2}}{b^{4} x^{2}} + \frac {2247 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{3}}{b^{6} x^{3}} + \frac {3843 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{4}}{b^{8} x^{4}} + \frac {3465 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{5}}{b^{10} x^{5}} + \frac {2625 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{6}}{b^{12} x^{6}} + \frac {945 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{7}}{b^{14} x^{7}} + \frac {315 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{8}}{b^{16} x^{8}} + 58\right )}}{315 \, a^{4} {\left (\frac {a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}}{b^{2} x} + 1\right )}^{9} {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 66, normalized size = 0.50 \begin {gather*} -\frac {\left (-b x +a \right ) \left (2 b^{3} x^{3}+12 a \,b^{2} x^{2}+33 a^{2} b x +58 a^{3}\right ) \sqrt {-b^{2} x^{2}+a^{2}}}{315 \left (b x +a \right )^{5} a^{4} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.45, size = 264, normalized size = 1.98 \begin {gather*} -\frac {2 \, \sqrt {-b^{2} x^{2} + a^{2}}}{9 \, {\left (b^{6} x^{5} + 5 \, a b^{5} x^{4} + 10 \, a^{2} b^{4} x^{3} + 10 \, a^{3} b^{3} x^{2} + 5 \, a^{4} b^{2} x + a^{5} b\right )}} + \frac {\sqrt {-b^{2} x^{2} + a^{2}}}{63 \, {\left (a b^{5} x^{4} + 4 \, a^{2} b^{4} x^{3} + 6 \, a^{3} b^{3} x^{2} + 4 \, a^{4} b^{2} x + a^{5} b\right )}} + \frac {\sqrt {-b^{2} x^{2} + a^{2}}}{105 \, {\left (a^{2} b^{4} x^{3} + 3 \, a^{3} b^{3} x^{2} + 3 \, a^{4} b^{2} x + a^{5} b\right )}} + \frac {2 \, \sqrt {-b^{2} x^{2} + a^{2}}}{315 \, {\left (a^{3} b^{3} x^{2} + 2 \, a^{4} b^{2} x + a^{5} b\right )}} + \frac {2 \, \sqrt {-b^{2} x^{2} + a^{2}}}{315 \, {\left (a^{4} b^{2} x + a^{5} b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.83, size = 143, normalized size = 1.08 \begin {gather*} \frac {\sqrt {a^2-b^2\,x^2}}{63\,a\,b\,{\left (a+b\,x\right )}^4}-\frac {2\,\sqrt {a^2-b^2\,x^2}}{9\,b\,{\left (a+b\,x\right )}^5}+\frac {\sqrt {a^2-b^2\,x^2}}{105\,a^2\,b\,{\left (a+b\,x\right )}^3}+\frac {2\,\sqrt {a^2-b^2\,x^2}}{315\,a^3\,b\,{\left (a+b\,x\right )}^2}+\frac {2\,\sqrt {a^2-b^2\,x^2}}{315\,a^4\,b\,\left (a+b\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- \left (- a + b x\right ) \left (a + b x\right )}}{\left (a + b x\right )^{6}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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